On estimation of the probability of absence of collisions of some random mappings

Let X and Y be finite sets and phi: (X, Y) --> be a mapping. Consider a ran dom mapping i --> phi(x(i), y(i)), where x(i) are arbitrarily chosen from t he set X, whereas (y(i)) is a random sample from Y without replacement. A t wo-sided bound is derived for the probability of absence of collisions of t his mapping. A case of mapping, defined as phi(x, y) = x + y modulo n, is c onsidered in particular. The results may be used in the selection of identi fication codes.